

where the notation \(k_i, k_h\in \mathcal {I}_{i,h}^{sf}\)  is equivalent to \(k_i\in \mathcal {I}_{i}^{sf}\)  and \(k_h\in \mathcal {I}_{h}^{sf}\) , similar arguments hold for \(k_i, k_j\in \mathcal {I}_{i,j}^{sf}\) ; and \(\alpha _{k_i}\in \mathbb {Z}\)  is used to avoid the existence of multiple cycles, \(\forall k_i\in \mathcal {I}_{i}^{sf}\)  and \(\forall i \in \mathcal {N}_{j}^f\cup \lbrace \nu \rbrace \) .
The first two constraints (REF )-() ensure that exactly one element of \(\Gamma _{i,s_if_i}\)  is intersected by \(\Gamma _j\) , \(\forall i\in \mathcal {N}_{j}^{f}\) .
The last constraint () and the definition of variables \(\lbrace \alpha _{k_i}\rbrace \)  ensure that all the waypoints \(m_{k_i}\)  and \(m_{k_h}\)  that satisfy \(\beta _{{k_i}{k_h}}\)  should belong to one big cycle where \(m_{\nu ,0}\)  is the last waypoint and is connected to \(m_{j,0}\) .
Simply speaking, assume that an additional cycle of waypoints (excluding \(m_{j,0}\) ) with length \(N_c>0\)  appears in \(\Gamma _j\) . Summing up the inequalities within () for all waypoints contained in that cycle would yield \(N_c\cdot (N_j^f+1)\le N_c\cdot N_j^f\) , leading to a contradiction. More details can be found in [1]}.
